Decision trees are graphical representations of a decision situation. They play a crucial role in both the representation and the analysis of decision problems. The main building blocks of a decision tree representation include (i) the decisions that need to be made (and the set of possible alternatives for each decision); (ii) the uncertainties present in the decision (and the set of possible outcomes for each uncertainty), and (iii) the set of possible consequences of the decision and their values (presented at the end of the tree). The decision tree is drawn from left to right in the chronological order of encountering the decisions, uncertainties and the possible consequences.
Using a decision tree (i) a decision is represented by a “decision object”, which is a square or a rectangle at the vertex of the object, from which branches representing the set of alternatives corresponding to that decision emanate; (ii) an uncertainty is represented by an “uncertainty object”, which is a circle or an oval at the vertex of the object, from which branches representing the possible outcomes of that uncertainty emanate, and (iii) the consequences of the decision, represented as end nodes of the tree, and given a triangular symbol. The decision tree is drawn from left to right to represent the chronological order of encountering the decisions, uncertainties and the possible consequences. An example of a decision tree is shown in FIG. 1. Decisions and uncertainties to the left precede (chronologically) decisions and uncertainties to the right. For example, the tree in FIG. 1 depicts a firm's decision of whether or not to introduce a new product, followed by an uncertainty about whether the competitor would introduce a competitive product in response, followed by a second decision that the firm might decide upon, which is the price for its product, and then finally an uncertainty about the competitor's price in response to their pricing decision.
The result of drawing the tree from left to right is that the size of each object (whether a decision object or an uncertainty object) is decided upon in advance before completing the tree. Note in FIG. 1 that the spacing between the branches in each object are equal, but the result is that the spacing between the consequences at the end of the tree is not uniform. Furthermore, with the current state-of-the-art, it is difficult to add more uncertainties and nodes to the end of the tree because the nodes will overlap. Furthermore, the tree is already drawn at this stage, so any attempts to increase the spacing will require redrawing the tree. But even if that is done, the congestion problem will persist as more decisions and uncertainties are added. This is a major limitation for the number of uncertainties and decisions that can be modeled and analyzed in a given decision situation.
Most complex decision situations we encounter today include multiple decision points and multiple uncertainties. Therefore it is important to have a method that can represent decision trees without congestion at the end of the tree, and that can incorporate any number of decisions and uncertainties.
Because of the advancements in scientific computing, decision trees have been embedded in numerous software packages and on many internet platforms. All of these software packages draw the tree in chronological order from left to right. Software packages that draw decision trees are known to have the following problems:                (i) The tree is often difficult to read because the branches are squeezed due to overlap and non-uniform spacing.        (ii) Branches and nodes overlap when the tree is scaled beyond a few decisions or uncertainties.        
Because of these reasons, it is difficult to model decisions using a decision tree beyond a few nodes when the tree is drawn in chronological order (from left to right), and the representation is not scalable. To overcome these problems, some software packages attempt to draw decision nodes and uncertainty nodes with very large spacing to minimize the overlap, but this makes the representation very inefficient in terms of space, particularly for large-size problems.
Because of the importance of decision trees in the analysis of decision problems, and because of the limitations in the current system of representation, there is an urgent need to provide an automated representation of decision trees that is both scalable (where the nodes do not overlap) and clear for users to read.
The proposed invention addresses this problem by drawing the tree at each stage from right to left (reverse chronological order) instead of from left to right (chronological order).
FIGS. 2-5 show examples of the proposed approach with uniform spacing at the end nodes by drawing the tree from right to left. The figures also show that the length of the horizontal branches in each stage of the tree is uniform, making the tree easy to read.